The (relativity) symmetry theoretical perspective for a theory
of particle dynamics works even better in the quantum cases
[1]}, [2]}. For the simple, `nonrelativistic', quantum
mechanics, starting with an irreducible representation of the
\(U(1)\)  central extension of the Galilean group, formulation
of all aspects of the dynamical theory is a natural consequence,
which highlights its parallel with the classical theory based on
the Hamiltonian dynamical picture [3]}, [4]}. We have
pointed out that the formulation gives an interesting, intuitive,
noncommutative geometrical picture of quantum physics
[1]}, [6]}, [7]}, [4]}, with the comprehensive commutative,
hence classical, limit retrievable as a contraction of the symmetry
and the representation. All that is only within the familiar symmetry,
including reference frame, transformation picture based on Lie
groups. Not considering a particle with nonzero spin, the irreducible
representation of the quantum relativity symmetry is essentially
unique. The representation space is just the familiar Hilbert space,
which is usually called the phase space for the quantum particle.
The projective Hilbert space as the space of rays in it is the exact
symplectic manifold each point of which corresponds to a distinct
physical state. While one can have a classical spatial translation,
it still has to be formulated as one of the phase space. With the
\(U(1)\)  central extension effectively incorporating the Heisenberg
commutation structure, the extended Galilean group simply
does not admit an invariant configuration/position space as
a representation. Just like the Minkowski spacetime cannot be
split into Lorentz invariant space and time parts, the quantum
phase space cannot be split into the configuration and the
momentum space parts, though both splitting work under
the Newtonian approximation. That is the reason for the
nonexistence of the notion of a quantum configuration space.
With the perspective that the `physical space' of a theory of
particle dynamics, or the proper model of it, can only be
a physically meaning notion as the space of all possible position
for a free particle, one should conclude that the only proper
model of the physical space as behind quantum mechanics is
the phase space [7]}. A quantum reference frame is then
always a reference frame for the latter, and hence a symmetry
transformation as coordinate transformation on the phase
space. The latter as a dual structure to the observable algebra
admits a noncommutative geometric picture with the position
and momentum observables as a basic set of noncommutative
canonical coordinates [6]}, [4]} for the phase space —
exactly as one would intuitively expect. That is the setting
to appreciate why a quantum reference frame transformations
as one we may want to think of as simply such a transformation
on the physical space like a simple spatial translation has to be
a canonical transformation and there is no way to look at it
independent of the phase space. There is no model of physical
space we can think of sensibly with only the position observables
as coordinates. The one with the position and momentum
observables as coordinates is exactly the right model of the
physical space as behind the theory of quantum mechanics, and
that intuitive quantum phase space agrees the space of physical
states as usually described as the (projective) Hilbert space of
infinite real/complex-number dimension. Each noncommutative
coordinate carries the information of infinite number of
real/complex-number coordinates, an explicit relation between
which is offer by a representation of each noncommutative
value of noncommutative coordinate as a sequence of infinite
number of complex numbers, like the one used above.
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